Can Lorentz force be expressed as a function of z in a static electromagnetic field?

[1Keenan]

Hi,

I have a particle moving in a static electromagnetic field in which E and B have the following components:
E=(E_x, 0, 0)
B=(-B_x, 0, 0)
and both depend on z, namely E_x(z) and B_x(z).
The particle is moving along z with constant velocity v=(0, 0, v_z).

If I want to express Lorentz force as a function of z, is it correct to write:

F(z)=q [(E *cross (1/v)) *cross (1/v) + B *cross (1/v)]?

I get this equation considering F=q(E+v *cross B) as a system of 6 differential equations:

dx/dt=0
dy/dt=0
dz/dt=v_z
dv_x/dt= q*E_x/m
dv_y/dt=q*B_x*v_z/m
dv_z/dt=0

and expressing them as a function of z

 

[tiny-tim]

Hi 1Keenan!

dx/dt=0
dy/dt=0

No.

If I want to express Lorentz force as a function of z, is it correct to write:

F(z)=q [(E *cross (1/v)) *cross (1/v) + B *cross (1/v)]?

(write "x" not "*cross" )

Where do the 1/v come from?

 

[1Keenan]

No.

Why?
My particle is moving along z and v=[0,0,v_z]
of course it will have a displacement along x and y but the intial velocity has only one component.
Could you please explain your point?

Where do the 1/v come from?

it comes from the differential equation:
dv_x/dt=qE_x/m
dv_y/dt=qB_xv_z/m

I change the variable t in z... it is a bit of algebra I can write you everything if you want so you can double check my manipulation.

 

[tiny-tim]

… the intial velocity has only one component.

but that doesn't mean that dx/dt = dy/dt = 0, not even initially

it comes from the differential equation:
dv_x/dt=qE_x/m
dv_y/dt=qB_xv_z/m

I change the variable t in z... it is a bit of algebra I can write you everything if you want so you can double check my manipulation.

i still don't get it

 

[1Keenan]

but that doesn't mean that dx/dt = dy/dt = 0, not even initially

What does it mean?

i still don't get it

How do you write it down?
I don't understand what is tricky for you...

 

[1Keenan]

I was thinking, and actually I'm doing something stupid, but I'm really interested in expressing those differential equation as function of z and I'm lost in papers full of my wrong formulas...
at the moment I'm not able to calculate 1+1... :(